Thank you for your post.
If you get to the Instant Web Publishing Home page, then the network is working correctly. Since the browsers are also using port 80 to access the internet, what occurs when the port is changed from 591 to port 80 (the default)?
From the symptoms you describe, it sounds like your local network is not always active. Otherwise, users would be able to log on and stay with the file. Besides FileMaker Pro running on your web server, are there any other web applications running? If so, do web users encounter the same problem?
Have you tried another machine?
Do you have access to another machine with a different static IP that is not on your local network? If so, can you try it there?
Any other information you can provide may be helpful in narrowing down the possible cause(s).